Frequency Response Function Estimation Research Articles

Finite element model updating using in-situ experimental data

Conventional model updating methods are based on frequency response function (FRF) and/or modal parameter estimates obtained from freely suspended, or sometimes rigidly constrained, sub-structures. These idealised boundary conditions are, however, often difficult to realise in a practical scenario. Furthermore, they are in conflict with the requirement that the sub-structure should also be measured whilst under a representative mounting condition. This paper addresses the question whether model updating can be achieved in the presence of an arbitrary or unknown boundary condition using in-situ measurements, i.e. without removing the sub-structure from its assembly. It is shown that some measurable properties, dynamic transfer stiffness and generalised transmissibility, are invariant to sub-structural boundary conditions and can therefore be obtained in-situ. It is further shown that, with minor adaption, existing transmissibility-based updating methods can be applied more widely than previously thought; to sub-structures whose boundary conditions are non-ideal. The theory is verified by a numerical beam example. Application to a resilient isolator is then demonstrated where a finite element model is successfully updated without removing the isolator from its assembly.

Frequency Response Function Measurements via Local Rational Modeling, Revisited

A finite measurement time is at the origin of transient (sometimes called leakage) errors in nonparametric frequency response function (FRF) estimation. If the FRF varies significantly over the frequency resolution of the experiment (= reciprocal of the measurement time), then these transients (leakage) errors cause important bias and variance errors in the FRF estimate. To decrease these errors, several local modeling techniques have been proposed in the literature. This article presents an overview of the existing methods and gives an in-depth bias and variance analysis of the FRF and disturbing noise variance estimates. In addition, a new local modeling approach is described, which combines the small bias error of the local rational approximation with the low noise sensitivity of the local polynomial approximation. It is based on an automatic local model-order selection procedure applied to a specific subclass of rational functions.

Forced vibration analysis of non-local strain gradient rod subjected to harmonic excitations

In this technical paper, forced longitudinal vibration response of non-local strain gradient (NLSG) rods is obtained and compared with that of non-local (NL), strain gradient (SG), and classical (CL) rods. To this end, size-dependent kinematics along with extended Hamilton’s principle are used to derive governing equations of motion. Classical clamped-free and non-classical clamped-forcing free-strained boundary conditions are adopted. For the first time, forced longitudinal vibratory behavior of non-classical and classical rods subjected to harmonic external loading is presented. In investigation process, effects of non-classical parameters over mechanical impedance, resonance frequency, mechanical frequency response function (FRF) (transmissibility), time period, and time-domain response are determined. Effect of non-classical parameters over time-domain response of the system is analyzed in details. Results are compared with benchmark approving good level of accuracy, disclosing the intensifying effects of material length scale parameter over natural frequency as well as mechanical impedance and resonance frequency. Meanwhile, non-local parameter leads to decrement of natural frequency, mechanical impedance and resonance frequency. Furthermore, it is realized that crest value of transmissibility is not predictable based on frequency and/or impedance variations with respect to non-classical parameters; and requires separate analysis for individual cases. In other words, a pattern referring to gradations of natural and resonance frequencies, mechanical impedance and non-classical parameters and rod models is detectable; however, such patterns do not hold for estimation of mechanical frequency response function (transmissibility peak value). Moreover, stiffness-softening and stiffness-hardening effects are the most impressive and influential factors over transmissibility peak values which can be highly important for energy harvesting rod analysis. Findings of this research point to the importance of proper non-classical theorems along with corresponding non-classical parameters to predict the response of small-scaled rods for wide range of applications in control of robotics, musical instruments, navigating equipment, space wheels, fly wheels, gyroscopes, sensors, actuators, resonators, vibration isolators and energy harvesters.

Nonlinear multiple breathing cracks detection using direct zeros estimation of higher-order frequency response function

On-board structures are currently submitted to high-level vibrations which may affect their integrity and operational lifetime. Structural health monitoring is then of great interest to avoid mechanical failures and to increase operational safety of mechanical systems. Indeed, in the case of cracked structures, the presence of cracks may affect the dynamic behavior. Thus, this paper focusses particularly on the detection and location of small cracks using the nonlinear response of cracked structures and more specifically using the zeros of the Higher-Order Frequency Response Function. The proposed method is based on the Direct Zeros Estimation algorithm to identify the antiresonances of the system. The locus of the identified antiresonances according to the crack location are discussed through numerical examples applied to the simplified nonlinear finite element model of a rectangular beam with multiple breathing cracks.

The study of seat vibration transmission of a high-speed train based on an improved MISO method

The field measurement was carried out on a high-speed train for the vibration transmission of the seat. The transmissibilities and coherences from the accelerations at the seat base to those on the seat pan and backrest were studied by an improved multi-input and single-output (MISO) system newly proposed by the authors. The improved method has the advantage of calculating the partial coherence independently of the sequences of the inputs and evaluating the contribution of the coherent parts between the inputs to the output compared with the existing method. It was found that although the high-speed train seat was exposed to complex vibrations from different sources, when the three inputs (vertical, lateral and roll accelerations) at the seat base were used as the inputs for MISO models, good multiple coherence functions could be obtained when studying the seat transmissibility to the vertical, lateral or roll vibration at the seat pan and backrest except in the frequency range where the anti-resonances of the inputs located. In such an exceptional case, the unsatisfactory multiple coherence could be explained by the anti-resonances of the input signals, which was further verified by the laboratory experiment. Then the on-site measurement was compared with the laboratory measurement in terms of the estimation of the coherence and transmissibility. Due to the advantages of the laboratory experiment, the contribution from the coherent parts is smaller, and the multiple coherence is higher than the field measurement. In the field measurement, that the inputs were coherent with the ‘noise’ was the main reason for the erroneous estimation of frequency response function (FRF), but the erroneous estimation of FRF resulting from the coherence between the inputs can to some degree be figured out by adopting MISO system. Finally, it was found the peaks in the seat transmissibilities caused by seat modes usually got reduced by the neighboring subject because of the damping brought from the subject.

Development and validation of experimental modal analysis with fixture-free oblique impact testing based on a vector projection method

Experimental modal analysis (EMA) with oblique excitation (i.e. oblique impact testing) is useful in improving the long test time problem of conventional EMA with normal excitation (i.e. triaxial normal impact testing), in order to extract all the important dynamic characteristics of a 3D complex structure. In this study, a new methodology involving a vector projection method is introduced to find the driving point frequency response function (FRF) in the oblique direction, without the need for a special fixture with an obliquely oriented impedance head. Hence, this presents a low cost and practical solution for scaling the mode shape, as compared to the traditional approach. Moreover, the concurrent forces characteristic of oblique excitation are used in the development of the theoretical relationship between FRFs with oblique excitation and those with normal excitation. This is important for the validation of the oblique impact testing result, such as the FRF and modal parameter estimations. The experimental results show that oblique impact testing has reliable and effective results, as compared with the triaxial normal impact testing, in terms of the FRF correlation, natural frequency discrepancy, modal damping ratio error and modal assurance criterion of the unit modal mass mode shape.

Fast extremum seeking using multisine dither and online complex curve fitting

Fast online optimization of uncertain Wiener systems using extremum seeking control (ESC) is investigated. Derivative estimation in extremum seeking is herefore described as an online parametric system identification problem. Multisine dithering is applied with frequencies around the first resonance frequency of the system to remove the time scale separation between dither and plant dynamics which is commonly required in ESC. Recursive use of the Fourier transform, over a moving window of historic data, provides a frequency response function estimate of the system’s local best linear approximation. Continuous online complex curve fitting is then applied to extrapolate to an estimate of the steady-state response which coincides with the local gradient of the steady-state objective function. An analysis of the closed-loop dynamics is provided. Transient improvements and robustness of the approach against plant variation are demonstrated with a simulation example.

A Unifying Theory of Driver Perception and Steering Control on Straight and Winding Roads

Novel driver support systems potentially enhance road safety by cooperating with the human driver. To optimize the design of emerging steering support systems, a profound understanding of driver steering behavior is required. This article proposes a new theory of driver steering, which unifies visual perception and control models. The theory is derived directly from measured steering data, without any a priori assumptions on driver inputs or control dynamics. Results of a human-in-the-loop simulator experiment are presented, in which drivers tracked the centerline of straight and winding roads. Multiloop frequency response function (FRF) estimates reveal how drivers use visual preview, lateral position feedback, and heading feedback for control. Classical control theory is used to model all three FRF estimates. The model has physically interpretable parameters, which indicate that drivers minimize the bearing angle to an “aim point” (located 0.25–0.75 s ahead) through simple compensatory control, both on straight and winding roads. The resulting unifying perception and control theory provides a new tool for rationalizing driver steering behavior, and for optimizing modern steering support systems.

Non-probabilistic method to consider uncertainties in frequency response function for vibration-based damage detection using Artificial Neural Network

Artificial neural network (ANN) has become a popular computational approach in the field of vibration-based damage detection, based on its ability to relate the nonlinear relationship between structural vibration characteristics and damage information. In the meantime, frequency response function (FRF) estimation has been proven effective as a dynamic parameter for damage detection due to its prevention of information leakage. In this regard, FRF is chosen as the input variable for ANN to detect structural damage in this study. However, the main concern in damage detection using FRF with ANN is the size of FRF data. A full-size FRF data will result in a wide composition range of the ANN input layer, thus affecting the iteration divergence in the network training process and resulting in the computational inefficiency. In most applications, principal component analysis (PCA) has been used to reduce the size of the FRF data before being fed to an ANN model. However, as the structures become more complex, the FRF data size also increases. The large size of FRF data may result in the PCA not being effective in selecting important information from the actual FRF data, leading to false damage detection. Moreover, the existence of uncertainties from modelling error and measurement error may also amplify the error in damage detection. Hence, this study proposes a combination of a non-probabilistic method with PCA to consider the problem of the existing uncertainties and the inefficiency of using FRF data in ANN-based damage detection. In this study, ANN is used to relate the FRF data to a damage feature. The input data for the network are the compressed real FRFs and the outputs are the elemental stiffness parameter (ESP). The compressed FRF data obtained from PCA provide a new damage index (DI) that is used as the input layer of the ANN. Based on the interval analysis method, the uncertainties in the new DI are considered to bind together to obtain the interval bound (lower and upper bounds) of the DI changes. The possibility of damage existence (PoDE) is designed to ascertain the relationship between the input and output parameters in the form of undamaged and damaged conditions. The verification conducted on a numerical model and a laboratory tested steel truss bridge model demonstrated that the proposed method is efficient in dealing with uncertainties using FRF for damage detection.

Modified Technique of Parameter Identification of a Permanent Magnet Synchronous Motor with PWM Inverter in the Presence of Dead-Time Effect and Measurement Noise

The paper considers the problem of parameter identification of the surface mounted permanent magnet synchronous motor (SPMSM) with pulse width modulated (PWM) inverter in the presence of dead time of power switches and other nonlinear distortions. Parameter identification of the SPMSM is required for the tuning of the torque control loop, because in some cases, the exact values of phase resistances and inductances are not known. In the absence of nonlinear disturbances, the problem of SPMSM parameters estimation is not difficult. The influence of the dead-time effect, back electromotive force and measurements noise introduces distortions in experimental output data sets, which leads to incorrect parameter estimation. Thus, there is a need to develop new designs of identification experiments and methods of processing of the experimental data. A detailed mathematical model of SPMSM with a PWM inverter in the presence of dead-time effect is considered in the paper. The negative influence of the dead-time effect on the results of parameter estimation is shown. A modified technique of parameter identification of SPMSM based on the estimation of frequency response function is proposed. The applied design of identification experiments, the type of excitation input signal, and methods of data processing allow us to minimize the influence of nonlinear disturbances and to reduce the variance of estimation of frequency response function. These features provide a high performance of SPMSM parameters estimation.

Large-Scale Regression: A Partition Analysis of the Least Squares Multisplitting

Indirect measurements of physical parameters of interest require a mathematical model in which these parameters are estimated from the gathered measurements. Within the least squares (LS) estimation, the parameters are estimated through a regression problem. The presence of dynamics, multiple sensors, and high sampling rates leads to high-dimensional regression matrices. This paper deals with solving such large-scale regression problems time efficiently. We revisit Renaut’s least squares multisplitting (LSMS) technique aimed at solving the ordinary LS problem in parallel. The LSMS decomposes the design matrix column-wise into several blocks. The global LS solution is subsequently replaced by an equivalent set of local LS problems that are to be solved in parallel. We study how the user should configure the partition of the multisplitting. We propose a partition design based on a clustering analysis and prove the consistency of this approach. The method is illustrated with dedicated numerical simulations for a highly scalable LS-based problem within engineering: frequency response function (FRF) estimation in the presence of missing output samples. Finally, its practical utility is shown with a laboratory measurement application.

A Performance Study of Controlled Impact Timing on Harmonics Reduction in Operational Modal Testing

As an alternative to operational modal analysis and classical experimental modal analysis (EMA), a novel method was introduced previously, namely impact-synchronous modal analysis (ISMA). The effectiveness ISMA on rotor and structural dynamic systems has been proven in previous literature. More recently, an automated impact device (AID) was introduced which utilized tachometer pulse as initiation signal and its effectiveness on ISMA was proven. An attempt to further enhance this device in term of equipment and cost is then proposed by replacing the tachometer with the in-use tri-axial accelerometer through utilizing the filtered response of cyclic load component as an initiation signal to control the impact device, which is also the primary aim for this study. Prior to modal testing, accuracy of this device is illustrated at desired phase angles of 0 deg, 90 deg, 180 deg, and 270 deg. Subsequently, frequency response function (FRF) estimations obtained for ISMA using enhanced AID has demonstrated the suppression capabilities of this device on disturbances, i.e., reduction of 93.58% at 30 Hz and 57.78% at 60 Hz, resulting in a high correlation for signature assurance criterion (SAC) and cross signature assurance criterion (CSAC). Modal parameters extracted from the EMA and ISMA using impact device are presented and compared, for the first three natural modes of the test rig. It is found that natural frequencies are deviating by less than 6%, whereas modal assurance criterion (MAC) values between the mode shapes of the two tests are found to be above 0.9.

Learning-based frequency response function estimation for nonlinear systems

ABSTRACTIn this paper, we perform the nonlinear frequency response function (FRF) estimation for a class of nonlinear systems. Two non-parametric estimation techniques are considered: radial basis function neural network (RBF-NN)-based estimation and support vector machine (SVM)-based estimation. Based on the system's available observations, the proposed estimation models are used to predict its frequency response. Simulation results are provided to demonstrate the model implementation. Finally, a comparative study is carried out to evaluate the effectiveness of the RBF-NN and SVM schemes, which has demonstrated that the SVM outperformed RBF-NN in the FRF estimation.

Automated impact device with non-synchronous impacts: a practical solution for modal testing during operation

Previous study has shown that synchronization of phases between impacts and the cyclic load component should be avoided to improve the effectiveness of operational modal testing, i.e. impact-synchronous modal analysis in obtaining a cleaner frequency response function (FRF) estimation with fewer number of averages. However, avoiding the phase synchronization effect is rarely achievable with the current manual impact hammer because of the lack of control of the impact timing. We investigate how to improve FRF estimation in the presence of harmonic disturbances, such as those present in operating rotating machines. An auto impact device is therefore introduced to replace the manual impact hammer. This device ensures that impact intervals can be applied at non-synchronous instances with respect to the harmonic disturbance. We demonstrate that this new device is a viable option for operational modal testing. It allows significant improvement in FRF estimation and shows good correlation of modal extraction data with benchmark experimental modal analysis results.

A Practical Estimation of Frequency Response Functions for System Decoupling Indirectly Using a Variable Cross Section Rod

It is of high importance to be able to decouple a system to obtain the dynamic characteristics of its substructures; however, the necessary frequency response functions (FRFs) of the coupling interface are usually challenging to measure due to the limited accessible space and complex geometries. In this paper, a measurement technique in the decoupling process of a coupled system is proposed in order to obtain the FRFs at coupling interface. Specifically, a variable cross section rod is adopted to transmit the dynamic behavior of coupling interface. The proposed technique has three advantages: (a) the thick end with large cross section can provide enough area for applying excitation force like using impact hammer and/or setting up sensors; (b) the slender end with small cross section can break through the spatial limitation more easily; and (c) the convenience that no additional experimental setup is required but just using an available variable cross section rod. Vibrational equation of the variable cross section probe method is derived and then combined with the existing decoupling theories. Finally, the proposed probe method and the new decoupling theory combining probe theory are validated through numerical simulations (FEM) and laboratory experiments, respectively. The results show its great practicability in decoupling process especially in low frequency range.

Comparison of experimental and operational modal analysis on a back to back planetary gear

Several modal analyse techniques are widely used to study the dynamic characteristics of a structure by identifying the modal parameters. This paper presents a comparison between an Experimental Modal Analysis (EMA) test, an Operational Modal Analysis (OMA) test and an Order Based Modal Analysis (OBMA) applied on a recirculating energy planetary gear. The OMA and OBMA offer a test of the planetary gear under its real boundary conditions, whereas the EMA is based on the frequency response function estimation. Using these different techniques, the back to back planetary gears modal parameters were identified. In a first step, the experimental results determined by EMA and OMA were correlated to the numerical results model obtained by a three-dimensional lumped parameter model. In a second step, the OBMA estimated modal parameters were compared to those of the EMA.According to the obtained results, OMA cannot excite all the modes. However, it was noticed that there is no significant difference between the modal parameters obtained by the EMA and the OBMA. The natural frequencies and damping ratios were deviating by 11% and 2.43%, respectively.

Error analysis and new dual-cosine window for estimating the sensor frequency response function from the step response data.

Aiming at reducing the estimation error of the sensor frequency response function (FRF) estimated by the commonly used window-based spectral estimation method, the error models of interpolation and transient errors are derived in the form of non-parameter models. Accordingly, window effects on the errors are analyzed and reveal that the commonly used hanning window leads to smaller interpolation error which can also be significantly eliminated by the cubic spline interpolation method when estimating the FRF from the step response data, and window with smaller front-end value can restrain more transient error. Thus, a new dual-cosine window with its non-zero discrete Fourier transform bins at -3, -1, 0, 1, and 3 is constructed for FRF estimation. Compared with the hanning window, the new dual-cosine window has the equivalent interpolation error suppression capability and better transient error suppression capability when estimating the FRF from the step response; specifically, it reduces the asymptotic property of the transient error from O(N-2) of the hanning window method to O(N-4) while only increases the uncertainty slightly (about 0.4 dB). Then, one direction of a wind tunnel strain gauge balance which is a high order, small damping, and non-minimum phase system is employed as the example for verifying the new dual-cosine window-based spectral estimation method. The model simulation result shows that the new dual-cosine window method is better than the hanning window method for FRF estimation, and compared with the Gans method and LPM method, it has the advantages of simple computation, less time consumption, and short data requirement; the actual data calculation result of the balance FRF is consistent to the simulation result. Thus, the new dual-cosine window is effective and practical for FRF estimation.

An accurate estimation of frequency response functions in output-only measurements

Frequency response functions (FRFs) can be estimated only from responses of structures without knowledge of input forces. Mode shapes identified from operational modal analysis (OMA) methods are unscaled, and the unscaled mode shapes provide unscaled FRFs. In this paper, mass change and mass–stiffness methods are employed to construct FRFs from the response-only measurements. A numerical case study of a cantilever beam is investigated using the finite element method. It is shown that the mass–stiffness change method provides more accurate results compared to the mass change method in the low-frequency range. A laboratory-scale steel beam is also tested using an OMA method and conventional hammer test. The experimental results show better accuracy of the identified FRFs in the low-frequency range when the mass–stiffness method is used.

Application of non-parametric methods for integrity monitoring of flight control systems

A method to monitor the integrity of flight control systems was developed and demonstrated. It is based on frequency response function estimates that are computed online with the transient local polynomial method. Confidence regions are derived and related to robust stability criteria to assess the probability of stability. Different from previous non-parametric approaches, the algorithm accounts for non-zero initial- and final conditions, colored noise and can be applied to open- and closed loop systems; when compared to parametric approaches, it is simpler, as no parametric model is required. The validity of the confidence regions is verified with Monte Carlo simulations. The usability to monitor changes of the plant dynamics is demonstrated by an example with changing dynamics.

Impact of the Missing Data Pattern, the Oversampling, the Noise Level, and the Excitation on Nonparametric Frequency Response Function Estimates

Nonparametric frequency response function estimation (FRF) is a first important step towards successful parametric modelling of the dynamics. In some applications such as, for example, low-cost wireless sensor networks, sensors are subject to failure (clipping, outliers) and the transmission errors of the wireless communication can be as high as 30%. Hence, nonparametric estimation of the FRF in the presence of missing data is an important issue. In this paper we study the impact of the missing data pattern, the missing data fraction, the oversampling (w.r.t. the bandwidth of the system), the signal-to-noise ratio and the type of excitation on the bias and variance of the FRF estimates.